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Re: [Orekit Developers] status on second order derivatives



Hi Evan,
Evan Ward <evan.ward@nrl.navy.mil> a écrit :


On 10/30/2014 07:10 AM, Luc Maisonobe wrote:
Hi Evan,

Le 29/10/2014 21:34, Evan Ward a écrit :
On 10/29/2014 03:14 PM, MAISONOBE Luc wrote:
Hi Evan,
Evan Ward <evan.ward@nrl.navy.mil> a écrit :

Hi Luc, Paul,

If I understand correctly this issue only affects propagators that do
not use Cartesian representation of the state. When a non-Cartesian
representation is used the propagator's position is combined with the
Keplerian velocity and the resulting combination is not locally tangent
(osculating) to the propagator's orbit. Am I understanding the issue
correctly?
Yes, this is exactly what happens.

Could this issue be solved by using the propagator's complete state to
determine the instantaneous P/V and use that to build a CartesianOrbit?
Not exactly. The numericalPropagator once only integrated
CartesianOrbit and it was considered a limitation. Since a few years,
we have added the possibility to directly integrate the orbit type
specifed by user. Moving back to CartesianOrbit only would be a
regression IMHO.

However what you suggest is also almost what I have in mind since we
really use the propagator complete state. In fact, for the numerical
propagator the conversion between acceleration and orbit is not done
at orbit level after integration, but rather at force level as we use
the Jacobian of the (P,V) vs orbital parameters conversion to directly
get parameters derivatives. When you look at it, the Gauss equations
are really the last columns of the Jacobian. This is done with the
TimeDerivatives interface, which for NumericalPropagator is
implemented using the Jacobian.

So as far as N?umericalPropagator is involved, I think consistency is
automatically preserved, since we do integrated something that is
computed from the Jacobian.

The DSST propagator on the other hand propagated directly in
equinoctial parameters and uses another dedicated implementation of
the TimeDerivatives interface.
I guess I don't understand why we need more than 6 parameters to specify
the osculating Keplerian orbit.
For Keplerian orbit, 6 parameters are enough and the relationship
between (a, e, i, pa, raan, M) and (x, y, z, vx, vy, vz) is a clear
bijection. If we restrict ourselves to Keplerian motion, this
relationship holds not only at t0 but also at t0+dt for every value we
want. If we use Keplerian parameters evolution (i.e. only M evolving)
and recompute at several steps the corresponding Cartesian, we will see
vx, vy, vz are really the derivatives or x, y, z, because the
relationship does take Kepler laws into account.

The problem arise when we introduce non-Keplerian parts. Then if we
blindly apply the same relationship for converting between the two sets
of parameters, we see that they are not consistent anymore.

What happens is that the mapping:

  (a, e, i, pa, raan, M)     <---> (x, y, z, vx, vy, vz)

is such that when da/dt = 0, de/dt = 0, ... dM/dt = n, then the computed
vx is dx/dt, vy is dy/dt and vz is dz/dt.

If now we decide to set arbitrarily da/dt to some made up non null value
and so on for all parameters, we still compute the same 6 values (x, y,
z, vx, vy, vz) as before (the initial value of a, e, i ... did not
change), but now differentiating the mapping shows that vx will not be
equal to dx/dt anymore, vy will not be dy/dt anymore and vz will not be
dz/dt anymore. The made up da/dt, de/dt will contribute to the results.
This is what happens for Eckstein-Hechler as it computes the left part
of the mapping and we (Orekit team) compute the right part using the
Keplerian-based mapping. If we want proper computation of vx, vy, vz, we
need the additional parameters.

I agree that we need to account for the element rate terms to correctly
compute the velocity. I think we could do the (a, e, i, pa, raan, M,
rates, other propagator state) --> (p, v) mapping within the Propagator
and then return a Orbit built from the correctly computed PV. This way
the returned orbit would be the osculating Keplerian orbit that matches
the propagator's position and velocity. If the user then translates the
osculating PV to osculating Keplerian elements they will be different
from the propagator's elements, but that is to be expected since the
propagator includes perturbations and osculating elements don't. (Maybe
this is the source of my confusion; when hear "osculating orbit" I think
of a conic orbit that matches the instantaneous PV at a single instant
in time.)

OK, I think this time I understood your concerns.
I'll give this a try, were the only modification to CircularOrbit will be to add a constructor that accepts both the circular and Cartesian parameters, hence the class will not do the conversion by itself anymore and only the propagator need to be aware of the circular derivatives.

My current setting was to use either circular or Cartesian but never build an orbit from both, so the conversion could be done only by the orbit class and therefore it should have known about the circular derivatives. With your proposal, it is simpler. Thanks for the suggestion.


The problem may not hold for NumericalPropagator (I still have to
check), because basically we do the computation the other way round. We
start from x, y, z, vx, vy, vz and deduce a, e, ... from the mapping. So
as long as our vx=dx/dt, vy=dy/dt, vz=dz/dt, the initial mapping will
still hold.

It seems that if we add rates to all the elements then the resulting
class could be classified as a Propagator.
It is only a way to get a consistent (P, V), and as a propagator it
would be really limited as it is a Taylor expansion. I would simply
qualify it as a perturbed orbit allowing local expansion.

Extending the Taylor series expansion to higher order derivatives makes
sense if we need to return the propagator's internal state, and the
satellite's PV in one consistent object. Is there a use case for
providing the propagator's internal state? I don't know, but I can't
think of any.

If we need to provide the user with acceleration as well as PV then I
agree that we need a bigger container class to hold the extra
information. Maybe the new PVACoordinates would be the right choice

I agree this is a good place. However, I did not create an extra class for this either, I simply added a third vector to PVCoordinates. We can discuss this later, when the merge of the branch will be attempted.

instead of a PerturbedOrbit class since I don't think we would get the
same extrapolation from a PerturbedOrbit stored in Keplerian vs
Cartesian elements. To put it another way, If I converted a
PerturbedKeplerianOrbit to a PerturbedCartesianOrbit would they both
follow the same path?

At second order level, in the neighborhood of the matching date, yes they should. Differences should appear at the first ignored derivative level and build up from there.


I'm not against the PerturbedOrbit classes; I just want to make sure we
don't choose a complex solution to a simple problem.

You are right. I have a clear tendency to over-engineer things, this is why I ask for advice from time to time. This is the great force of collaborative development : people don't get stuck in their own errors too long.

So to be clear, here is the consensus :

 - I will simplify what I have done to limit as much as possible
   the changes to CircularOrbit.
 - In this setting, orbits will not know about non-Keplerian effects.
 - As this knowledge is needed for accurate and more importantly
   consistent velocity computation, the mapping between orbital parameters
   and Cartesian parameters will be precomputed by the propagators
 - If the mapping is not precomputed and orbits are built from one
   set of parameters only, then the mapping will be Keplerian only
   and will be performed byt orbit classes just as it is now

best regards,
Luc


There might be a case for
this intermediate level of fidelity/speed, but is it worth the added
complexity? Especially since we already have one that is fast (the Orbit
class) and one that is precise (the Propagator classes). Maybe I'm still
not understanding your proposal.
I detected the problem when checking some pointing attitude modes that
are related to spacecraft velocity (alignment of the spacecraft axis
with ground drift for Earth observation). This mode transforms something
that is a derivative (velocity) into a regular non-differentiated value
(an angle). So we go up one order of derivation. For such modes, a wrong
velocity induces a wrong angle and a wrong acceleration induces a wrong
angular rate. As the angular rate is used for time shift, I needed it to
be accurate.

You also mentioned issues with Doppler which would also occur.

We do need accurate velocity, and we do need a velocity that is
consistent with the derivative of the position.

Definitely agree that getting the velocity correct is important. Thanks
again for all the work you've put into solving this bug.

Best Regards,
Evan


best regards,
Luc

Best Regards,
Evan

Then we wouldn't have to modify the existing set of Orbit classes, and
the user would see the correct osculating P/V. (This might be equivalent
to your second approach.)

As far as where to put the code, it seems like the conversion code would
be specific to the internal state representation used by the propagator,
so I think it makes sense to keep the code as private to the propagator.
Though if you think there would be other uses for the conversion, I
think a public static factory method would work well.
Yes, the conversion code is propagator dependent. The first
implementation I played with is Eckstein-Hechler propagator, and it
definitely is Eckstein-Hechler specific (it's a simple derivation of
the original equations, so each time we did compute something like e =
a * c + b * d, now we also have another statement to compute eDot =
aDot * c + a * dCot + bDot * d + b * dDot). For numerical propagator
we already have the derivatives since we start from the derivatives
and integrate them, so its even simpler. For DSST, this will be a mix
as the mean elements are integrated and the short periodics terms are
computed from Fourier coefficients which are straightforward to
differentiate. For ephemeris-based propagator, we will need to compute
the derivatives of the underlying polynomials, which is also
straightforward.

Thanks Luc for finding this issue and doing the analysis. I can see how
this would be an issue when computing the Doppler as well as time
shifting.
your welcome

best regards,
Luc

Best Regards,
Evan

On 10/29/2014 06:57 AM, MAISONOBE Luc wrote:
Hi Paul,

paulcefo <paulcefo@buffalo.edu> a écrit :

Luc,

Do I correctly understand that your concern is that Keplerian
transformations do apply outside the osculating space?
The problem I had was that we did use Keplerian-only expression
to set up local Taylor expansions around the current point (a few
seconds away). This was slightly wrong when all the parameters were
time-dependent and not only the anomaly was time-dependent. Of course,
the error increasing with the time offset with respect to the central
date at which the Taylor expansion is built. The fix was simply to
not forget the derivatives of these other parameters.

This Taylor expansion feature is a built-in feature available in all
Orekit orbits, it typically allows to do computation in the vicinity of
an already computed point without needed to trigger a complete
propagator.
It can even be used for some computation inside the run of a
propagator,
as for example when the higher level propagator takes care of the long
term propagation and at each step we need some additional points
surrounding the current step to compute attitude evolution in some
specific modes.

My concern was how to implement this fix in our current architecture,
and more precisely were to put the code: in an existing class or in
a dedicated class which would be used only by propagators.

best regards,
Luc

Paul

--
Dr. Paul J. Cefola
Consultant in Aerospace Systems, Spaceflight Mechanics, &
Astrodynamics
Adjunct Faculty, Dept. of Mechanical and Aerospace Engineering,
University at Buffalo (SUNY)

4 Moonstone Way
Vineyard Haven, MA 02568
USA

508-696-1884 (phone on Martha's Vineyard)
978-201-1393 (cell)

paulcefo@buffalo.edu
paul.cefola@gmail.com

On 10/29/2014 6:02 am, MAISONOBE Luc wrote:
Hello,

As some of you may be aware, I have been working for a few months on
second order derivatives in the git branch
position-velocity-acceleration. This work is still ongoing but I hope
to finish it for 7.0 and merge the branch back to master soon. For
now, there are still failing tests so I can't do it.

This change should allow us to reach several goals :

- improved accuracy in shiftedBy methods
- improved accuracy in interpolators (with user-defined
  choices to use or not first and second derivatives
  from the sample)
- improved accuracy in attitude
- removal of ugly hidden finite differences in some classes
  (most notably attitude modes) with hard-coded steps
- hopefully faster Earth transforms, by replacing Hermite
  interpolation with single point extrapolation
- availability of non-Keplerian acceleration everywhere
- availability of angular acceleration in attitude and frames
- proper composition of dynamics in frames
- possibility to propagate orbits in non-inertial frames
- possibility to propagate orbits without a central body
  (interplanetary missions, Lagrange point missions, ...)

There is one point that bothers me right now. As I removed some of
the
ugly finite differences, some non-regression tests started to fail. I
finally found the raw cause of these failures and was surprised to
discover an old bug in the way we use the osculating orbits produced
by the Eckstein-Hechler analytical propagator. This propagator takes
zonal terms into account, and produces directly circular
parameters a,
ex, ey, ... When we compute anything related to geometry, we compute
Cartesian coordinates using the Orbit getPVCoordinates method. As the
Orbit classes do not know anything about the perturbation, the (P, V)
pair does in fact implicitly relies on Keplerian-only expressions. So
the velocity part is *not* consistent with the derivative of the
position. The real derivative of the position takes the non-Keplerian
effects into account which are ignored by getPVCoordinates. The
difference is small, but as the tests threshold were deliberatly very
tight, the tests started to fail when the various pointing directions
were not computed anymore from finite differences mainly involving
position and when they relied on the computed velocity. So the
problem
already happens in the master branch, it is not specific to the
introduction of acceleration (it was just detected here during
testing).

The solution is in fact quite simple. If an orbit has been
produced by
a non-Keplerian propagator, the propagator already knows about the
derivatives of the orbital elements (which are circular in the
Eckstein-Hechler model case but can be any kind of parameters for
other propagators). The propagator should therefore provide these
derivatives to the orbit so they can be used in the PVCoordinates
conversion. The code is very simple and straightforward. I have
checked this and got very interesting results with
Eckstein-Hechler/Circular, as for example a simple interpolation over
a 900s arc with proper velocity/acceleration has a 88m error with two
base points now whereas it was 5162 m before (and 0.02m vs 650m for 3
points, 1.0e-5m vs 259m for 4 points).

Here is what bothers me:

Should we create specialized classes for perturbed orbits or
should we
simply add a constructor to the existing orbits with the parameters
derivatives and set them to 0 when they are not known?

For my tests, I created PerturbedCircularOrbit which extends
CircularOrbit and override the protected initPVCoordinates method and
the public shiftedBy and interpolate methods. I could also have
simply
moved everything into CircularOrbit with a new constructor.

I do not like much the PerturbedXxxxOrbit approach, as it forces to
create also additional entries in the OrbitType enum with additional
converters and it becomes awkward if for example a user configures a
NumericalPropagator to generate XxxxOrbit, despite this propagator
will in fact really generate PerturbedXxxOrbit because it is what a
Numerical propagator is for. So there should be either an internal
modification of the user setting from OrbitType.XXXX to
OrbitType.PERTURBED_XXXX or an error triggered which would invalidate
*all* current user code as it would become forbiddent to generate
XXXX
orbits now.

On the other hand, the drawback of modifying the existing classes to
hold the non-Keplerian derivatives is that they will consume more
memory. I don't think it is a problem with current computers.

In any case, initial orbits created directly from user code or by
reading files would not include the derivatives and therefore will be
built as usual (by calling the unmodified classes in the first
approach, or by using the already existing constructors in the second
approach, assuming these constructors will automatically set the
derivatives to Keplerian-only values). In any case, full-blown
perturbed orbits will be created internally by Orekit propagators,
which can easily be modified to provide the derivatives they know (by
creating instances of the new derived classes in the first approach,
or by using new constructors with additional parameters in the second
approach).

My humble opinion would be to use the second approach to solve this
bug. I will probably do this in the position-velocity-acceleration
branch so it will include accelerations right from the start and will
be merged to master at the same time as the rest of the branch. Of
course, this will be a dedicated commits (Git branches are great!).

What do you think ?

best regards,
Luc

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